1. Introduction to Rotational Dynamics

Rotational Dynamics is the study of how forces cause rotational motion.

Just as Newton’s Second Law describes translational motion:

$$\sum F=ma$$

Rotational Dynamics uses the rotational equivalent:

$$\sum \tau =I\alpha$$

where:

• $\tau$ = torque
• $I$ = moment of inertia
• $\alpha$ = angular acceleration

This equation is one of the most important relationships in AP Physics C Mechanics.

2. What Causes Rotation?

A force can produce:

• translation
• rotation
• both translation and rotation

Whether a force causes rotation depends on where the force is applied relative to the axis of rotation.

For example:

• pushing a door near its hinges produces little rotation
• pushing the same door near its handle produces much more rotation

The quantity that measures rotational effectiveness is torque.

3. Torque

Definition

Torque is the rotational equivalent of force.

It measures the tendency of a force to cause rotation.

Vector form:

τ⃗=r⃗×F⃗\vec{\tau} = \vec{r} \times \vec{F}τ=r×F

where:

• r⃗\vec{r}r = position vector from the axis to the point of application
• F⃗\vec{F}F = applied force

Magnitude of Torque

$$\tau =rF\sin \theta$$

where:

• $r$ = distance from the axis
• $F$ = force magnitude
• $\theta$ = angle between r⃗\vec rr and F⃗\vec FF

4. Maximum and Minimum Torque

Maximum Torque

Occurs when:

θ=90∘\theta=90^\circθ=90∘

Then:

$$\tau =rF$$

The force is perpendicular to the radius.

Zero Torque

Occurs when:

θ=0∘\theta=0^\circθ=0∘

or

θ=180∘\theta=180^\circθ=180∘

The force acts through the axis.

Then:

$$\tau =0$$

5. Direction of Torque

Torque is a vector.

Its direction is determined by the right-hand rule.

Right-Hand Rule

1. Point your fingers along r⃗\vec rr.
2. Curl them toward F⃗\vec FF.
3. Your thumb indicates the direction of τ⃗\vec\tauτ.

Common convention:

• Counterclockwise → positive
• Clockwise → negative

6. Moment of Inertia

Definition

The Moment of Inertia measures an object’s resistance to angular acceleration.

It is the rotational equivalent of mass.

Symbol:

$$I$$

Units:

kg⋅m2\text{kg}\cdot\text{m}^2kg⋅m2

7. Point Mass Moment of Inertia

For a point mass:

I=mr2I=mr^2I=mr2

where:

• $m$ = mass
• $r$ = distance from the axis

Mass farther from the axis contributes more strongly because of the squared distance.

8. Continuous Objects

For extended bodies:

I=∫r2 dmI=\int r^2\,dmI=∫r2dm

This calculus expression is used to derive moments of inertia for rods, disks, rings, and spheres.

9. Common Moments of Inertia

Thin Hoop or Ring

I=MR2I=MR^2I=MR2

Solid Disk

I=12MR2I=\frac12 MR^2I=21​MR2

Solid Cylinder

I=12MR2I=\frac12 MR^2I=21​MR2

Solid Sphere

I=25MR2I=\frac25 MR^2I=52​MR2

Thin Rod About Center

I=112ML2I=\frac1{12}ML^2I=121​ML2

Thin Rod About End

I=13ML2I=\frac13 ML^2I=31​ML2

10. Newton’s Second Law for Rotation

The rotational form of Newton’s Second Law is:

$$\sum \tau =I\alpha$$

This equation plays the same role in rotational motion that:

$$\sum F=ma$$

plays in translational motion.

11. Rotational Analogy Table

Understanding this analogy simplifies rotational mechanics.

12. Example Problem

A disk has:

I=2 kg⋅m2I=2\,\text{kg}\cdot\text{m}^2I=2kg⋅m2

A torque of:

$$8\text{N}\cdot \text{m}$$

is applied.

Find the angular acceleration.

Solution

Using:

$$\sum \tau =I\alpha$$ $$8=(2)\alpha$$ α=4 rad/s2\alpha=4\,\text{rad/s}^2α=4rad/s2

13. Rotational Equilibrium

An object is in rotational equilibrium when:

$$\sum \tau =0$$

In this situation:

$$\alpha =0$$

The object:

• remains at rest
• or rotates with constant angular velocity

14. Combined Translation and Rotation

Many real objects both translate and rotate.

Examples:

• rolling wheels
• bowling balls
• cylinders on ramps

In such problems:

$$\sum F=ma$$

and

$$\sum \tau =I\alpha$$

must often be applied simultaneously.

15. Common AP Physics C Mistakes

Mistake 1

Using force instead of perpendicular force.

Only the perpendicular component contributes to torque.

Mistake 2

Ignoring the distance from the axis.

Torque depends on the lever arm:

$$\tau =rF\sin \theta$$

Mistake 3

Confusing mass and moment of inertia.

Objects with identical masses can have different moments of inertia depending on how the mass is distributed.

Summary

Torque:

$$\tau =rF\sin \theta$$

Moment of inertia:

I=mr2I=mr^2I=mr2

for a point mass.

Rotational Newton’s Second Law:

$$\sum \tau =I\alpha$$

Key ideas:

• torque causes rotational acceleration
• moment of inertia resists rotational acceleration
• rotational dynamics is the rotational version of Newton’s Laws
• rotational equilibrium occurs when net torque is zero

Rotational Dynamics provides the foundation for understanding rotating systems and leads directly to the study of rotational energy, rolling motion, and angular momentum in AP Physics C Mechanics.